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Theorem lubfun 17586
Description: The LUB is a function. (Contributed by NM, 9-Sep-2018.)
Hypothesis
Ref Expression
lubfun.u 𝑈 = (lub‘𝐾)
Assertion
Ref Expression
lubfun Fun 𝑈

Proof of Theorem lubfun
Dummy variables 𝑥 𝑠 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 funmpt 6366 . . . 4 Fun (𝑠 ∈ 𝒫 (Base‘𝐾) ↦ (𝑥 ∈ (Base‘𝐾)(∀𝑦𝑠 𝑦(le‘𝐾)𝑥 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦𝑠 𝑦(le‘𝐾)𝑧𝑥(le‘𝐾)𝑧))))
2 funres 6370 . . . 4 (Fun (𝑠 ∈ 𝒫 (Base‘𝐾) ↦ (𝑥 ∈ (Base‘𝐾)(∀𝑦𝑠 𝑦(le‘𝐾)𝑥 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦𝑠 𝑦(le‘𝐾)𝑧𝑥(le‘𝐾)𝑧)))) → Fun ((𝑠 ∈ 𝒫 (Base‘𝐾) ↦ (𝑥 ∈ (Base‘𝐾)(∀𝑦𝑠 𝑦(le‘𝐾)𝑥 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦𝑠 𝑦(le‘𝐾)𝑧𝑥(le‘𝐾)𝑧)))) ↾ {𝑠 ∣ ∃!𝑥 ∈ (Base‘𝐾)(∀𝑦𝑠 𝑦(le‘𝐾)𝑥 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦𝑠 𝑦(le‘𝐾)𝑧𝑥(le‘𝐾)𝑧))}))
31, 2ax-mp 5 . . 3 Fun ((𝑠 ∈ 𝒫 (Base‘𝐾) ↦ (𝑥 ∈ (Base‘𝐾)(∀𝑦𝑠 𝑦(le‘𝐾)𝑥 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦𝑠 𝑦(le‘𝐾)𝑧𝑥(le‘𝐾)𝑧)))) ↾ {𝑠 ∣ ∃!𝑥 ∈ (Base‘𝐾)(∀𝑦𝑠 𝑦(le‘𝐾)𝑥 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦𝑠 𝑦(le‘𝐾)𝑧𝑥(le‘𝐾)𝑧))})
4 eqid 2801 . . . . 5 (Base‘𝐾) = (Base‘𝐾)
5 eqid 2801 . . . . 5 (le‘𝐾) = (le‘𝐾)
6 lubfun.u . . . . 5 𝑈 = (lub‘𝐾)
7 biid 264 . . . . 5 ((∀𝑦𝑠 𝑦(le‘𝐾)𝑥 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦𝑠 𝑦(le‘𝐾)𝑧𝑥(le‘𝐾)𝑧)) ↔ (∀𝑦𝑠 𝑦(le‘𝐾)𝑥 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦𝑠 𝑦(le‘𝐾)𝑧𝑥(le‘𝐾)𝑧)))
8 id 22 . . . . 5 (𝐾 ∈ V → 𝐾 ∈ V)
94, 5, 6, 7, 8lubfval 17584 . . . 4 (𝐾 ∈ V → 𝑈 = ((𝑠 ∈ 𝒫 (Base‘𝐾) ↦ (𝑥 ∈ (Base‘𝐾)(∀𝑦𝑠 𝑦(le‘𝐾)𝑥 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦𝑠 𝑦(le‘𝐾)𝑧𝑥(le‘𝐾)𝑧)))) ↾ {𝑠 ∣ ∃!𝑥 ∈ (Base‘𝐾)(∀𝑦𝑠 𝑦(le‘𝐾)𝑥 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦𝑠 𝑦(le‘𝐾)𝑧𝑥(le‘𝐾)𝑧))}))
109funeqd 6350 . . 3 (𝐾 ∈ V → (Fun 𝑈 ↔ Fun ((𝑠 ∈ 𝒫 (Base‘𝐾) ↦ (𝑥 ∈ (Base‘𝐾)(∀𝑦𝑠 𝑦(le‘𝐾)𝑥 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦𝑠 𝑦(le‘𝐾)𝑧𝑥(le‘𝐾)𝑧)))) ↾ {𝑠 ∣ ∃!𝑥 ∈ (Base‘𝐾)(∀𝑦𝑠 𝑦(le‘𝐾)𝑥 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦𝑠 𝑦(le‘𝐾)𝑧𝑥(le‘𝐾)𝑧))})))
113, 10mpbiri 261 . 2 (𝐾 ∈ V → Fun 𝑈)
12 fun0 6393 . . 3 Fun ∅
13 fvprc 6642 . . . . 5 𝐾 ∈ V → (lub‘𝐾) = ∅)
146, 13syl5eq 2848 . . . 4 𝐾 ∈ V → 𝑈 = ∅)
1514funeqd 6350 . . 3 𝐾 ∈ V → (Fun 𝑈 ↔ Fun ∅))
1612, 15mpbiri 261 . 2 𝐾 ∈ V → Fun 𝑈)
1711, 16pm2.61i 185 1 Fun 𝑈
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 399   = wceq 1538  wcel 2112  {cab 2779  wral 3109  ∃!wreu 3111  Vcvv 3444  c0 4246  𝒫 cpw 4500   class class class wbr 5033  cmpt 5113  cres 5525  Fun wfun 6322  cfv 6328  crio 7096  Basecbs 16479  lecple 16568  lubclub 17548
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-reu 3116  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-riota 7097  df-lub 17580
This theorem is referenced by:  joinfval  17607  joinfval2  17608
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